Tom Stoppard’s "Rosencrantz and Guildenstern Are Dead" opens with a puzzling scene in which the title characters are betting on coin throws and observe a seemingly astonishing run of 92 heads in a row. Guildenstern grows uneasy and proposes a number of unsettling explanations for what is occurring. Then, in a sudden change of heart, he appears to suggest that there is nothing surprising about what they are witnessing, and nothing that needs any explanation. He says ‘…each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.’

In this article I argue that Guildenstern is right – there is nothing surprising about throwing 92 heads in a row. I go on to consider the relationship between surprise, probability and belief.

The above paper claims that there’s nothing “surprising” about a fair coin coming up heads 92 times in a row, basically on the grounds that any random string HTTHHHTTTHTHT ... is equally probable.

While I agree with the latter claim, I don’t agree with conclusion drawn from it. In fact, it’s preposterous, and this is the sort of philosophy paper – and it won a prize – that brings “philosophy of X” into contempt amongst practitioners of X – in this case mathematics, or statistics – not that I count myself as one of those.

There are faint resonances with ancient discussions I’ve had with friends on the question of the probabilities of unusual events – eg of miracles – where I know our views differed. This is a less contentious case, I think. Here, the probability of the event is agreed upon and has a calculable numerical value, which is not the case with the probabilities of miracles. However, it is relevant to their perceived credibility.

it’s important when responding to a “preposterous” argument to get to the bottom of it and find out exactly what’s wrong with it. My suspicions are that – irrespective of psychology and how our brains might be wired – we need to invoke things like the central limit theorem3, information theory4, analogies with the Sorites5 Paradox, the Lottery Paradox6 and the like.

The author (Martin Smith) has written lots of (fairly contentious) stuff on testimony versus probabilities, which I’ve collected and am looking forward to studying.

Detailed Arguments

Examples of “Surprising” Events→ The light not coming on when I flick the switch→ A colleague promising to attend a meeting and “no showing”→ My car is not where I left it

Smith aims to argue – in contrast – that throwing 92 heads would not be surprising.

The above is a normative claim – we might well be surprised, but – Smith will argue – we ought not to be. → This claim – it substantiated – has far-reaching consequences for what we should believe, given our limited evidence, in other circumstances.

The Conjunction Principle: what is the surprisingness-rating of (e1 & e2) given the surprisingness-ratings of the individual events?

The surprisingness of the conjunction two unsurprising events would also be unsurprising – Smith claims – if the two events are unconnected (like two successive tosses of a fair coin).

If the conjunction of the two events were impossible, then the surprisingness of the conjunction occurring would be extreme.

So, Smith claims this conjunction principle: If it’s unsurprising for event e1 to happen, and it’s unsurprising for event e2 to happen, and these two events are independent of one another, then it’s unsurprising for e1 and e2 to both happen.

Smith now posits that the conjunction principle can be iterated so that 92 consecutive Hs are not surprising, given that no individual H is surprising.

Two earlier attempts to define the conjunction principle are addressed:-→ George Shackle7 (1950s-60s): has, for two independent events, Sup (e1 & e2) = max {Sup (e1), Sup (e2)}, where Sup in [0 , 1]. → Wolfgang Spohn8 (1980s): has, for two independent events, Sup (e1 & e2) = sum {Sup (e1), Sup (e2)}, where Sup in Z+. For both Shackle’s “mathematical theory of surprise” and Spohn’s “ranking theory”, a completely unsurprising event – like a single coin-toss resulting in H – has Sup of 0, so the conjunction of two such events – and indeed any number thereof – also has Sup of 0, ie. is completely unsurprising.

Surprise versus Unlikeliness

But, isn’t 92 consecutive Hs rather unlikely? Yes, its probability is 2 x 10-27; near miraculous, and – many think – very surprising.

Smith quotes some injudicious remarks by a trio of mathematical greats9: d’Alembert in 1760s, Cournot in the 1840s and Borel in 1942 to the effect that such low-probability events never happen (“Borel’s Law”). Smith asks whether such a claim – while an exaggeration – might be approximately true – such events very rarely happen, and are therefore surprising.

Smith thinks the claim – far from being near the truth – is almost the exact opposite. His argument is that we can have a situation where every outcome is highly improbable but not one where every outcome is surprising. This shows that surprise and probability come apart. The reason for this is that in the coin-tossing case, any outcome is just as unlikely as any other; so, HTTHHTHT … is just as unlikely as HHHHHHHH … . For 92 coin tosses, each outcome – one of which is bound to happen – is a 1-in-5,000-trillion-trillion event; so – when a 1-in-5,000-trillion-trillion event does happen, it should not be surprising.

Smith tries to reinforce this by appeals to other improbable but unsurprising events – like the precise temporal, volumetric and molecular dimensions of a breath; these – in their exact measurements – are even more unlikely than the coin-toss sequences. Similarly, the precise time my phone rings. We don’t want to be in a perpetual state of surprise. A vague claim – a phone-call “this weekend” – can be likely, but a specific one will be unlikely.

Smith sums up by saying that the conjunction principle has allowed him to prove that 92 Hs is not surprising, even though it is extremely unlikely.

Expectation

Smith admits that 92 consecutive Hs are not to be expected where expectation is used in the mathematical sense of the probability weighted average of the possible values of a random variable. He notes that the probabilities of the number of heads in 92 tosses approximates to a normal distribution, with the bell curve peaking at 46 and 92 Hs more than 9 standard deviations from the mean. Smith asks whether this is one of the occasions where an extreme divergence from the mean ought to elicit surprise.

Smith has an interesting argument against the tempting conclusion that surprise is appropriate. He notes that there’s a 73.8% chance that there will be between 40 and 50 Hs out of 92 tosses. He makes an analogy between this and the claim that “my phone will ring over the weekend”. The reason either claim is so likely is that there are so many – individually improbable – ways for the claim to be satisfied.

So, why is this not a conclusive argument in favour of surprise – very mild in this case – being appropriate if expectation isn’t met?

The question that naturally springs to mind at this stage is, what are the chances of all four players being dealt a complete suit? Well, there are 4! or 24 ways in which each player can receive a complete suit, and division by 24 leaves us with10 odds of 2,235,197,406,895,366,368,301,559,99911 to 1 against. If the entire adult population of the world were to play bridge in every waking moment for ten million years, it would still be ten million to one against one of these perfect deals turning up.

So how can we account for all the newspaper reports of four players in a bridge game each receiving a complete suit? The answer is invariably a joker, not in the pack but amongst the players or, more probably, in the ranks of the kibitzers. It is not too hard to switch a pack without being spotted.

Smith references Decision Order and Time in Human Affairs, 2nd ed. (Cambridge University Press, 1969) [Contains Shackle’s most detailed presentation of his ideas about surprise. His axioms, and his struggles over axiom 7, can be found in chapter X.]

Smith references The Laws of Belief (Oxford University Press, 2012) [Spohn’s definitive presentation of ranking theory and its various applications. Discusses Shackle and surprise in section 11.1. The law of conjunction for negative ranks is principle 5.16 in chapter 5.]

This book is referenced by Wikipedia, and Spohn’s work looks worth following up!